Entropy Production along a Stochastic Trajectory and an Integral Fluctuation Theorem

For stochastic nonequilibrium dynamics like a Langevin equation for a colloidal particle or a master equation for discrete states, entropy production along a single trajectory is studied. It involves both genuine particle entropy and entropy production in the surrounding medium. The integrated sum of both $\Delta s_{\mathrm{tot}}$ is shown to obey a fluctuation theorem $⟨\exp [-\Delta s_{\mathrm{tot}}]⟩=1$ for arbitrary initial conditions and arbitrary time-dependent driving over a finite time interval.